English

Only finitely many Tribonacci Diophantine triples exist

Number Theory 2016-01-21 v2

Abstract

Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1u<v<w1\le u<v<w such that uv+1,uw+1,vw+1uv+1,uw+1,vw+1 are Tribonacci numbers. The proof depends on the Subspace theorem.

Keywords

Cite

@article{arxiv.1508.07760,
  title  = {Only finitely many Tribonacci Diophantine triples exist},
  author = {Clemens Fuchs and Christoph Hutle and Nurettin Irmak and Florian Luca and Laszlo Szalay},
  journal= {arXiv preprint arXiv:1508.07760},
  year   = {2016}
}

Comments

This is a preprint of the Materials accepted for publication in "Mathematica Slovaca"

R2 v1 2026-06-22T10:45:04.507Z